Ultrametric tree

Figure 2. An illustration of the ultrametric tree formed by locally optimal paths from the apex to the base of a 1 + 1 dimensionaJ delta. Three non-intersecting paths are drawn to schematically represent the directed optim ll paths from O to the sites A, A2 and A3. The ultrametric distances between these three sites on the base have the relation U(Ai,A2) = U(A3,Ai) > U(A2, A3) which is the same as the relation in Eq. (22).

Fig. 7.3 A representation of the (conjectured) ultrametric distribution of spin-glass equilibrium states. The leaves of the tree at bottom are identified with the states overlaps between states are measured by the number of levels it takes to trace the states back to their common roots . For the three states a, 0 and 7, for example, we have that qot y = q y = q and = 92 > 9l-...

The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. 

The following constitutes the definition of the distance between two points in an ultrametric space. The points in an ultrametric space on a given hierarchical level are the ends of the Cayley tree branches (Fig. 13). The number of points on the rath level of the Cayley tree is equal to Nn = j". Each point on the nth level can be numbered ... 

The points of the discrete ultrametric space (Cayley tree junctions) on the nth level, namely, N , are divided into clusters (groups). Each cluster contains j points the distance between which is / = 1 and has its progenitor on the (.n — l)th level. The number of such clusters is N /j = jn l ... 

This approximate equation means that the ultrametric space has a logarithmic metric. Thus, when constructing a fractal set, each element corresponds to a point of the ultrametric space with geometric image represented by the Cayley tree. 

The hierarchical chain of changes from the initial state (t = 0) to the final one (t —> oo) can be compared to a Cayley tree [25,26] (see Fig. 64). Here, the knots of the Cayley tree will correspond to static ensembles a and p which correspond to the dots in ultrametric space divided by the distance lap. 

The value of lap is defined by the number of steps over the levels of the Cayley tree up to the mutual knot in Fig. 64 and it yields the extent of a hierarchical link. Therefore, both the barrier height, Qap, and the relaxation time, xap, are connected with functions of the distance lap in ultrametric space, that is,... 

In the previous discussion we have broadly compared trees generated from widely differing character sets. For each tree, a matrix of ultrametric distance values has been computed in order to test the goodness of fit of the cluster analyses to the data (Rohlf and Sokal, 1981). The resultant coefficients of cophenetic correlation indicated very good fits (r > 0.9) for all tested trees with the exception of the tree based on body... 

The ultrametricity of the tree is a direct consequence of the non-intersecting property of the locally optimal paths. In ultrametric space, any three points A, A2 and A3 satisfy the inequality [22] ... 

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